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# The value of $a$ for which the volume of the parallelopiped formed by vectors $\hat i+a\hat j+\hat k,\:\:\hat j+a\hat k\:\:and\:\: a\hat i+\hat k$ becomes minimum is ?

$\begin{array}{1 1} \frac{1}{\sqrt 3} \\ \frac{1}{\sqrt 3} \\ \sqrt 3 \\ - \sqrt 3 \end{array}$

Can you answer this question?

Volume of a parallelopiped = $\left|\begin {array}{ccc}1 &a & 1\\0 &1 &a\\a & 0 & 1\end {array}\right |$
$=1+a(a^2-1)$
Given: Volume is minimum.
$\therefore\:\large\frac{dV}{da}$$=0\:and\:\large\frac{d^2V}{da^2}$$=- ve$.
$\Rightarrow\:3a^2-1=0,$ $i.e.,\:a=\pm\large\frac{1}{\sqrt 3}$
and
$\large\frac{d^2V}{da^2}$$=6a$ which is -ve if $a=-\large\frac{1}{\sqrt 3}$
and is +ve if $a=\large\frac{1}{\sqrt 3}$
$\therefore\: a=\large\frac{1}{\sqrt 3}$ when Volume is minimum.
answered Nov 12, 2013